A long structural member subject to a compressive load is called a strut.
Struts with large cross sections compared with the length generally fail under
compressive stress and the conventional failure criteria apply. When the cross
section area is not large compared to the length i.e the member is slender, then the
member will generally fail by buckling well before the compressive yield strength
is reached.

The notes below relate to uniform straight members made from homogeneous engineering
materials used within the elastic operating range. It is assumed that an end load is
applied along the centroid of the ends. The strut will remain straight until
the end load reaches a critical value and buckling will be initiated. Any increase in load will result in a
catastrophic collapse and a reduction in load will allow the strut to straighten.
The value of the critical load depends upon the slenderness ratio and the end fixing
conditions. The slenderness ratio (λ )is defined as the effective length =Le / the
least radius of gyration = k of the section The principal end fixing conditions are
listed below

Note: The derivation below is based on a strut with pinned ends. A similar method can be used
to arrive at the Euler loads for other end arrangements which will confirm the basis for the factors in arriving at the
equivalent length b.

The lowest value of W resulting from this procedure is called the Euler load
(We ) and failure of long slender beams due to buckling results from this
much earlier than failure due pure compression.
As the moment of inertia I = A.k 2 and the end force W = σ A. The formula
can be rewritten

Important Note: The value of I and the equivalent value of k are assumed to be the minimum values for the section
under consideration

Validity of Eulers theory

This theory takes no account of the compressive stress. For a metal with a compressive
strength of less than 300 N/mm2 and a Young's Modulus of about 200 kN/mm2.
The strut will tend to fail in compression if the slenderness ratio (Le/ k) is less than 80.
Therefore for steel Eulers equation is not reliable for slenderness ratios less than
80 and really should not be used for slenderness ratios less than 120.

Rankine - Gordon Criteria

This criteria is based on experimental results.

This criteria suggests that the strut will fail at a load given by.

1 / W R = 1 / Wc + 1 / We

Wc = Compressive failure Load
We = Euler Load

Substituting c = σc / ( π2 E) - A constant for each material

This design criteria provides more accurate buckling loads than the euler theory
especially at lower slenderness ratios. At higher slenderness ratios
the two methods yield similar results. The experimental values for c are not in direct
agreement with the theoretical values. BS 449-2:1969 includes tables for the safe working stresses for
all slenderness ratios and a range of steel specifications.

Table showing approximate values of c

Material

c

Mild Steel

1/7500

Wrought Iron

1/9000

Cast Iron

1/1600

Wood

1/3000

Perry Robertson formula (BS 449-2 )

Important ..The notes and equation and table below is provided for general guidance. For detail structural
design it is important to refer to the identified standards. The information below is
only a trivial relative to the level of detail provided in the standard.

The equation below is used as the basis for the allowable design stresses as
provided in the relevant tables in BS 449 and is considered the most reliable of the methods available
for buckling loads for long slender struts..The equation below is similar to that provided in appendix B of BS 449 part 2 :1969